
the other values of the ranks and the numbers of abelian
crowns are zero. We note that if \(1\le i<j\le 3\) , two chief factors of \(S\)  corresponding to chief factors of \(\hat{G}_i\)  and \(\hat{G}_j\) , respectively, cannot be \(S\) -isomorphic, because the groups of automorphisms induced by \(S\)  on the chief factors in the primitive groups \(G_i\)  and \(G_j\)  are not isomorphic. Therefore \(\operatorname{cr}^{\mathfrak {A}}_3(S)=1\) , \(\operatorname{cr}^{\mathfrak {A}}_7(S)=9\) , \(\operatorname{cr}^{\mathfrak {A}}_8(S)=146\) , \(\operatorname{r}_{\operatorname{GL}_3(2)}(S)=57\) ; the other
values are zero (we note that, in fact, \(S\)  coincides with the direct product \(\hat{G}_1\times \hat{G}_2\times \hat{G}_3\) ). Since the indices of the maximal subgroups of
\(\operatorname{GL}_3(2)\)  are 7 and 8 (see for instance
[1]}), we conclude that \(\operatorname{rs}_7(S)=\operatorname{rs}_8(S)=57\)  and
\(\operatorname{ro}_{168}(S)=\operatorname{rm}_{168}(S)=57\) . The crowns of chief factors of order 8 in \(\hat{G}_1\)  are minimal normal subgroups, in \(\hat{G}_2\)  are products of three minimal normal subgroups, while in \(\hat{G}_3\)  are products of two minimal normal subgroups. The crowns of chief factors of order 7 in \(\hat{G}_1\)  and \(\hat{G}_2\)  coincide with the corresponding chief factors. There is a unique crown composed of two central chief factors of order 3. Hence \(S\)  has \(16+16\cdot 3+114\cdot 2=292\)  chief factors of order 8, 8 chief factors of order 7, 2 chief factors of order 3, and 57 chief factors isomorphic to \(\operatorname{GL}_3(2)\) .
